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Solve Linear Inequalities

Solve Linear Inequalities

Author: Sophia Tutorial
Description:

Solve a linear inequality.

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Tutorial
what's covered
  1. Graphing Inequalities
  2. Solving Linear Inequalities

1. Graphing Inequalities

Solving inequalities is very similar to solving equations with one exception. if we multiply or divide by a negative number, the symbol will need to flip directions. We will keep that in mind as we solve inequalities.

hint
When multiplying or dividing by a negative number, the inequality sign switches. For example, greater than becomes less than, and less than becomes greater than.


2. Solving Linear Inequalities

short dash 2 x greater-than or slanted equal to 6
Divide both sides by short dash 2
stack space short dash 2 space with bar on top space stack space short dash 2 space with bar on top
Divide by a negative - flip symbol!
x less-than or slanted equal to short dash 3
Graph, starting at short dash 3, going down with ] for less than or equal to



File:9684-ineq1.png
open parentheses short dash infinity comma space short dash 3 close parentheses
Interval Notation

The inequality we solve can get as complex as the linear equations we solved. We will use all the same patterns to solve these inequalities as we did for solving equations. Just remember that any time we multiply or divide by a negative the symbol switches directions (multiplying or dividing by a positive does not change the symbol!)

Solve and give Interval notation

3 open parentheses 2 x minus 4 close parentheses plus 4 x space less than space 4 open parentheses 3 x minus 7 close parentheses plus 8
Distribute
6 x minus 12 plus 4 x space less than space 12 x minus 28 plus 8
Combine like terms
10 x minus 12 space less than space 12 x minus 20
Move variable to one side
stack negative 10 x space space space space space space space space space space space space space minus 10 x with bar below
Subtract 10x from both sides
short dash 12 space less than space 2 x minus 20
Add 20 to both sides
stack plus 20 space space space space space space space space space space space space space space plus 20 with bar below

8 space less than space 2 x
Divide both sides by 2
stack space 2 space with bar on top space space space space space stack space 2 space with bar on top

4 space less than space x
Be careful with graph, x is larger!



File:9683-ineq2.png
open parentheses 4 comma infinity close parentheses
Interval Notation

summary
When graphing inequalities, it is important to be careful when the inequality is written backwards as in the above example (4 less than x rather than x greater than 4). Often students draw their graphs the wrong way when this is the case. The inequality symbol opens to the variable, this means the variable is greater than 4. So we must shade above the 4.
Solving linear inequalities is similar to solving equations, except that we use an inequality symbol instead of an equal sign. When we're solving an inequality and you multiply or divide by a negative number, your inequality symbol is going to switch directions. Also, when we're solving a compound inequality, any operation done between the inequality symbols must be done on the other side of both inequality symbols.

Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html